Open games (Ghani, Hedges, Winschel & Zahn) give game theory a compositional syntax in which a game becomes a morphism wired to others in sequence and parallel, and the monoidal product of players’ selection functions is exactly the equilibrium condition.
A game’s content is a player’s rationality, packaged as a selection function. Given a way to score outcomes, a selection function returns the choices it considers optimal, whereas equilibrium is the property these wirings preserve. The question is how selection functions combine.
Let
that fuses two players into a joint one. This fusion is the equilibrium condition.
Theorem. The monoidal product is the Nash condition
For selection functions built from
, the fused device selects the pair iff is a pure-strategy Nash equilibrium of the two-player game with payoff . Compositional structure and game-theoretic solution concept coincide, since taking the monoidal product of selection functions computes exactly the best-response fixed point.
Definition. Assembling the game category
Applying the monoidal Grothendieck construction to the indexed monoidal
turns it into a single strong monoidal functor . Its total category has as objects the pairs (object of , selection function on it), and the monoidal product on carries the laxator . Wiring games together and combining their equilibria are therefore one and the same operation.
References
- N. Ghani, J. Hedges, V. Winschel, P. Zahn, Compositional Game Theory (LICS 2018)